Upload
breana-muriel
View
218
Download
0
Tags:
Embed Size (px)
Citation preview
Graduate Program in Business Information Systems
Inventory Decisions with Uncertain Factors
Aslı Sencer
BIS 517- Aslı Sencer 2
Uncertainties in real life
Demand is usually uncertain.
Probability distributions are used to represent uncertain factors.
Ex: Demand is normally distributed OR
Demand is either 20,30,40 with respective probabilities 0.2, 0.5, 0.3.
BIS 517- Aslı Sencer 3
Stochastic versus Deterministic Models Mathematical models involving probability are
referred to as stochastic models. Deterministic models are limited in scope
since they do not involve uncertain factors. But they are used to develop insight!
Stochastic models are based on “expected values”, i.e. the long run average of all possible outcomes!
BIS 517- Aslı Sencer 4
Example: Drugstore
A drugstore stocks Fortunes.They sell each for $3 and unit cost is $2.10. Unsold copies are returned for $.70 credit. There are four levels of demand possible. How many copies of Fortune should be stocked in October?
Payoff Table:
Demand
Event Probability
ACTS
Q = 20 Q = 21 Q = 22 Q = 23D = 20 .2 $18.00 $16.60 $15.20 $13.80
D = 21 .4 8.00 18.90 17.50 16.10
D = 22 .3 8.00 18.90 19.80 18.40
D = 23 .1 8.00 18.90 19.80 20.70
BIS 517- Aslı Sencer 5
Solution:
The expected payoffs are computed for each possible order quantity:
Q = 20 Q = 21 Q = 22 Q = 23$18.00 $18.44 $17.90 $16.79
Optimal stocking level, Q*=21 at an optimal expected profit of $18.44
If the probabilities were long-run frequencies, then doing so would maximize long-run profit.
BIS 517- Aslı Sencer 6
Example: Drugstore Payoff Table(Figure 16-1)
123456789
101112131415161718
A B C D E F
PROBLEM: Fortune Magazine
Act 1 Act 2 Act 3 Act 4Events Probability Q = 20 Q = 21 Q = 22 Q = 23
1 D = 20 0.2 $18.00 $16.60 $15.20 $13.802 D = 21 0.4 $18.00 $18.90 $17.50 $16.103 D = 22 0.3 $18.00 $18.90 $19.80 $18.404 D = 23 0.1 $18.00 $18.90 $19.80 $20.70
Act 1 Act 2 Act 3 Act 4Q = 20 Q = 21 Q = 22 Q = 23
Expected Payoff $18.00 $18.44 $17.96 $16.79
PAYOFF TABLE EVALUATION
Problem Data
Act Summary
18C
=SUMPRODUCT($B$9:$B$12,C9:C12)
BIS 517- Aslı Sencer 7
Single-Period Inventory Decision:The Newsvendor Problem Single period problem (periodic review) Demand is uncertain (stochastic) No fixed ordering cost Instead of h ($/$/period) we have hE ($/unit/period=ch) Instead of p ($/unit) we have pS and pR-c
Q: Order Quantity (decision variable)D: Demand Quantity
Costs:c = Unit procurement costhE = Additional cost of each item held at end of inventory cycle = unit inventory holding cost-salvage value to the supplierpS = Penalty for each item short (loss of customer goodwill)pR = Selling price
BIS 517- Aslı Sencer 8
Modeling the Newsvendor Problem
The objective is to minimize total expected cost, which can be simplified as:
where is the expected demand.
QDifQ
QDifDSales
)()()( QBcppQBQchcQTEC RSE
QdifQdcppcQ
QdifdQhcQQTC
RS
E
)(
BIS 517- Aslı Sencer 9
Optimal order quantity of the Newsboy Problem
Q* is the smallest possible demand such that
chcpp
cppQD
ERS
RS
]*Pr[
BIS 517- Aslı Sencer 10
Example: Newsboy Problem
A newsvendor sells Wall Street Journals. She loses pS = $.02 in future profits each time a customer wants to buy a paper when out of stock. They sell for pR = $.23 and cost c = $.20. Unsold copies cost hE = $.01 to dispose. Demands between 45 and 55 are equally likely. How many should she stock?
BIS 517- Aslı Sencer 11
Example: Solution
Discrete Uniform Distribution
Demand is either 45,46,47,..., 55 each with a probability of 1/11.
192
2001202302202302
......
...chcpp
cpp
ERS
RS
P(D<=Q*)=0.2 Q*=47 units.
12
Newsvendor Problem (Figure 16-3)
1234567
8
910111213141516171819202122232425262728293031
A B C D E F G
PROBLEM: Wall Street Journal
Parameter Values:Cost per Item Procured: c = 0.20Additional Cost for Each Leftover Item Held: hE = 0.01
Penalty for Each Item Short: pS = 0.02
Selling Price per Unit: pR = 0.23Number of demands for probability distribution = 11
Optimal Values:Optimal Order Quantity: Q* = 47Expected Demand: mu = 49.5Total Expected Cost: TEC(Q*) = $10.07Expected Shortages: B(Q*) = 2.66Probability of Shortage: P[D>Q*] = 0.80
Cumulative Number ofDemand Probability Probability shortages
45 0.05 0.05 0.046 0.06 0.11 0.047 0.09 0.20 0.048 0.12 0.32 1.049 0.17 0.49 2.050 0.20 0.69 3.051 0.12 0.81 4.052 0.08 0.89 5.053 0.06 0.95 6.054 0.04 0.99 7.055 0.01 1.00 8.0
SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM
BIS 517- Aslı Sencer
BIS 517- Aslı Sencer 13
Multiperiod Inventory Policies When demand is uncertain, multiperiod inventory
might look like this over time.
BIS 517- Aslı Sencer 14
Multiperiod Inventory Policies
The multiperiod decisions involve two variables: Order quantity Q Reorder point r
The following parameters apply: A = mean annual demand rate k = ordering cost c = unit procurement cost ps = cost of short item (no matter how long) h = annual holding cost per dollar value = mean lead-time demand
BIS 517- Aslı Sencer 15
Multiperiod Inventory Policies: Discrete Lead-Time Demand
The following is used to compute the expected shortage per inventory cycle:
The following is used to compute the total annual expected cost:
rd
L dDrdrB Pr
rBQA
prQ
hckQA
Q,rTEC S
2
BIS 517- Aslı Sencer 16
Multiperiod Inventory Policies: Discrete Lead-Time Demand Solution Algorithm.
Calculate the starting order quantity:
Determine the reorder point r*:
Determine optimal order quantity:
This procedure continues –using the last Q to obtain r and r to obtain the next Q- until no values change.
hcAk
Q2
1
Ap
hcQrD
S
1*Pr
hc
rBpkA*Q S 2
BIS 517- Aslı Sencer 17
Example:
Annual demand for printer cartridges costing c = $1.50 is A = 1,500. Ordering cost is k = $5 and holding cost is $.12 per dollar per year. Shortage cost is pS = $.12, no matter how long. Lead-time demand
has the following distribution. Find the optimal inventory policy.
BIS 517- Aslı Sencer 18
Example: Solution
The starting order quantity is:
r* = 7 cartridges. B(7) = (8–7)(.03) + (9–7)(.01) + (10–7)(.01)=.08 and
the optimal order quantity is:
28950112
5500121
..,
Q
93.
500,1)5.0(
2895.1)12.0(11*Pr
Ap
hcQrD
S
29050112
0850550012 ..
..,*Q
BIS 517- Aslı Sencer 19
Example: Solution (cont’d.)
Q=290 leads to r=7, so the solution is optimal.The optimal inventory policy is:
r* = 7 Q* = 290
Optimal annual expected cost is:
71.52$08.0290
15005.047
2
290)5.1)(12.0(5
290
1500TEC(7,290)
20
Multiperiod Discrete BackorderingIteration 1
12345678910111213141516171819202122
2324252627282930313233
A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Printer Cartridges
Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5Number of demands for probability distribution = 11
Optimal Values:Optimal Order Quantity: Q* = 288.68Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.7094$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05
Cumulative Number ofDemand Probability Probability Shortages
0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 210 0.01 1.00 3
14
1516
171819
G=SQRT((2*G7*G6)/(G9*G8))
=INDEX(C23:C42,MATCH(LOOKUP(1-((G9*G8*G14)/(G10*G7)),E23:E42,C23:C42),C23:C42,0)+1)=SUMPRODUCT(C23:C42,D23:D42)
=(G7/G14)*G6+G9*G8*(G14/2+G15-G16)+G10*(G7/G14)*G18=SUMPRODUCT(D23:D42,F23:F42)=1-VLOOKUP(G15,C23:E42,3)
BIS 517- Aslı Sencer
21
123456789
10
111213141516171819202122
2324252627282930313233
A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Printer Cartridges
Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5
Number of demands for probability distribution = 11
Optimal Values:Optimal Order Quantity: Q* = 290Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.71$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05
Cumulative Number ofDemand Probability Probability Shortages
0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 2
10 0.01 1.00 3
14
1516
171819
G=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))
=INDEX(C23:C42,MATCH(LOOKUP(1-((G9*G8*G14)/(G10*G7)),E23:E42,C23:C42),C23:C42,0)+1)=SUMPRODUCT(C23:C42,D23:D42)
=(G7/G14)*G6+G9*G8*(G14/2+G15-G16)+G10*(G7/G14)*G18=SUMPRODUCT(D23:D42,F23:F42)=1-VLOOKUP(G15,C23:E42,3)
Multiperiod Discrete BackorderingIteration 10
14G
=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))
BIS 517- Aslı Sencer
22
12345678910111213
14151617181920212223
A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Printer Cartridges
Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5Number of demands for probability distribution = 11
Iteration, i Qi ri B(ri) TEC(Qi,ri)
1 289 7 0.08 52.71$ 2 290 7 0.08 52.71$ 3 290 7 0.08 52.71$ 4 290 7 0.08 52.71$ 5 290 7 0.08 52.71$ 6 290 7 0.08 52.71$ 7 290 7 0.08 52.71$ 8 290 7 0.08 52.71$ 9 290 7 0.08 52.71$ 10 290 7 0.08 52.71$
Multiperiod Discrete BackorderingSummary
BIS 517- Aslı Sencer